Exploring Nanoscale Structure in Perovskite Precursor Solutions Using Neutron and Light Scattering

Tailoring the solution chemistry of metal halide perovskites requires a detailed understanding of precursor aggregation and coordination. In this work, we use various scattering techniques, including dynamic light scattering (DLS), small angle neutron scattering (SANS), and spin–echo SANS (SESANS) to probe the nanostructures from 1 nm to 10 μm within two different lead-halide perovskite solution inks (MAPbI3 and a triple-cation mixed-halide perovskite). We find that DLS can misrepresent the size distribution of the colloidal dispersion and use SANS/SESANS to confirm that these perovskite solutions are mostly comprised of 1–2 nm-sized particles. We further conclude that if there are larger colloids present, their concentration must be <0.005% of the total dispersion volume. With SANS, we apply a simple fitting model for two component microemulsions (Teubner–Strey), demonstrating this as a potential method to investigate the structure, chemical composition, and colloidal stability of perovskite solutions, and we here show that MAPbI3 solutions age more drastically than triple cation solutions.

In conventional dynamic light scattering, as used in this study, a laser is directed into the colloidal suspension and the intensity of the scattered light is detected at a set angle. This will contain a combination of constructive and destructive interferences as the light randomly scatters off colloids in the dispersion. This interference pattern changes in intensity as the particles move due to Brownian motion which in turn depends on the particle size as well as the temperature and viscosity of the solvent.
Larger particles diffuse more slowly. 2,3 DLS measures the fluctuations in the interference pattern over time, this process is shown schematically in Figure S1. The output of the measurement is an autocorrelation function showing the fluctuation of intensity over time called G2 1 . G2 can be related to the electric field autocorrelation function G1 which measures the averages motion of particles relative to one another.
Where B is a baseline constant, τ is the lag time between time (t) and (t+ τ) and β is factor associated with the measuring equipment. For a polydisperse system, G1 can be written as: Here, is a decay constant that is directly related to the diffusion constant of the colloids in dispersion over time increment ) . Whereas, q is the scattering vector (which incorporates the refractive index of the solvent 2 ). The hydrodynamic radius Rh of a molecule can be determined through the Stokes-Einstein equation Where T is temperature at which the measurements are taken, 0 is the Boltzmann constant and is the viscosity of the solvent. Through the above equations, a series of exponential decays can be fitted to G2 for each particle population to determine Rh. As can be seen in Figure S1, the exponential decay G2 will take longer for larger colloids. 3 Additionally, the more extended the decay becomes, the greater the polydispersity of the colloids in the dispersion.
For particles much smaller than the probe lasers wavelength (d< λ/10), light is scattered uniformly according to Rayleigh scattering theory. However, for particles comparable in size to λ the scattering can be described by Mie scattering theory, which depends on the refractive index of the scattering colloid and solvent, as well as the shape of the colloid, resulting in a pronounced angular dependence to the scattering 2 .
The intensity of Rayleigh scattering (I) is proportional to the square of the volume of the colloid and thus to its radius (r) as This means that a particle with = 50 nm will scatter with a million times the intensity of a particle with a radius of = 5 nm. This is a serious consideration when studying multimodal particle dispersions, as the scattering from a few larger particles will dominate that from many more but smaller particles.
Mie theory can be used to convert data from this intensity-weighted domain into either a numberweighted or, as here, a volume-weighted domain.
For a solution containing Na particles of size a, and Nb particles of size b, the magnitude of the intensity peak at size a is evaluated using: 4 Using the Rayleigh approximation, in which mass is proportional to r 3 -the volume percentage of colloid a can be estimated using: In this work, spherical geometry is assumed for the conversion from intensity to volume. and these are averaged to provide the results in Figure 1a.

Film Calculations
From the volume-weighted DLS size distributions, we assume a Gaussian distribution around a mean particle size and use the area underneath this curve to find the Vol% of a particular particle size as shown in Figure S3.
These data represent the hydrodynamic diameter of these particles therefore the particle radius is half this. If we assume these are spherical colloids, then the volume of a large and small particle is 4.1x10 -13 cm -3 and 2.6x10 -21 cm -3 , respectively. The total solute vol% of these TC solutions is roughly 14% and from the DLS we can tell that of this 0.0000235% is made up of large particles. Therefore, in any volume of TC solution, 3.291x10 -6 % of that will made up of large particles. So given the volume of a particle and how much volume they all occupy, we can estimate how many there are. Following through, we find that there are 8.11x10 6 large particles in 1ml. In a wet film of average thickness 10µm, this would lead to 8150 large particles/cm 2 . By the same logic, we calculate that there are 5.4x10 19 small particles in 1ml and therefore 5.78x10 16 small particles/cm 2 in the same wet film.
6 Figure S4. a Volume Percentage Distribution from DLS measurements on TC 2. Integrated percentages measured as the area beneath the curve. b. Schematic of the "wet film" used in the DLS

Small-Angle Neutron Scattering
Small angle neutron scattering (SANS) is a powerful technique for probing the average size, shape, distribution, and number of scattering entities in a sample. In a SANS experiment, a collimated beam of neutrons with wave vector 5 ;;;⃗ is directed towards a sample and neutrons scattered with wave vector 6 ;;;;⃗ at an angle 2θ are collected by a 2D detector, positioned at a distance L from the sample ( Figure S5).
Typically, the measured 2D scattering pattern is radially integrated to generate a 1D intensity profile as a function of the scattering vector ⃗ , defined as the difference between the wave vectors of the incident and scattered neutron beams: where is the neutron wavelength calculated from de Broglie's equation ( =  (1) and (2), the magnitude of the scattering vector is given by:

Equation S11
This is the same expression for q used in DLS, except neutron refractive indices are ~1.
The measurable -range in a SANS experiment is therefore dependent on the wavelength of the neutrons used and the distance between the detector and the sample. Placing the detector far away from the sample allows small angles to be probed relating to low-values. Scattering at low-q arises from large length scales d in the system, according to the reciprocal relationship between real-space and -space: 8 There are many models to describe small angle scattering from different types of nanostructure. Here, the Teubner-Strey model is used to characterise the perovskite precursor solutions. The Teubner-Strey model assumes a two phase system 4 . In this system, the solvent and solutes are the majority and minority phases respectively. The scattering intensity profile over a given -space can be described using the following equations: Depending on the domain size d, correlation length, ξ, volume fraction, 1 , and respective scattering length densities, 1/4 . In neutron scattering the phase problem means that d can be the domain size or periodicity between particles. We here argue that the d from the Teubner-Strey model corresponds to the average domain size of any solute phase and ξ is the average length scale before a phase change.
The scattering length densities (SLDs) and volume fraction of the solvent and solute particles can also be extracted using this model. To explore what this could represent in our systems, we refer to Figure   S6. Here, we can see that if phase B (the solute) is formed of large aggregates (as seen in S6a), both the domain size d and average distance between phases (ξ (= Σξn/n)) is larger than for a dispersed solution of nanoparticles (Fig. S5b). The solute material representing the minority phase within the Teubner-Strey model is shown in with dark blue circles, with the solvent (i.e., the majority phase) shown in light blue. Insets show the average spacing between "phase changes" for each system, ξn, which is averaged to give the correlation length ξ (= Σ ξn/n). The domain size of the minority phase is shown for each case as d.
There are several benefits for using neutron scattering to study perovskite solutions compared to conventional small angle X-ray scattering (SAXS) methods. SAXS experiments on perovskite solutions often suffer from low transmission rates as the solutions are very concentrated and Pb has a large scattering cross-section. SANS overcomes this due to the highly penetrative, weakly interacting nature 9 of neutrons. In contrast to SAXS, which relies on the electromagnetic interaction between the incident X-ray, and differences in electron density of the sample, neutrons are an uncharged probe so only interact with nuclei within the sample. Scattering arises when there is a difference in neutron scattering length densities (SLD) between the atoms, molecules or ions of the solvent and solute in the sample.
The neutron SLD of a molecule is equivalent to the summation of nuclear scattering lengths > of each constituent atom divided by the molecular volume 9 :

Equation S14
Neutron scattering lengths do not vary linearly across the periodic table as for X-ray scattering lengths  Table S1.  Table S1. A summary of chemical formulae, mass densities and neutron scattering length densities (SLDs) of the solvents, perovskite compositions and respective perovskite precursor materials used in this work. * Scattering length densities determined using 7 . ** Organic material densities estimated using 8 .   Figure S12. A schematic depicting the basic structure of how a SESANS instrument works. 9

Spin-Echo Small-Angle Neutron Scattering
In a spin-echo SANS (SESANS) instrument the scattering angle is encoded using the neutron spin, this is similar to NMR where 1 H nuclear can be measured by exploiting the spin-1 2 T property of the 1 H nuclei and using Larmor precession in magnetic fields. As neutrons are also spin 1/2 particles, we can measure the scattering angle by the use of carefully shaped magnetic fields. The key here is the fact that any accumulated precession is proportional to the strength of the magnetic field and the path traversed by the neutron through the field. In a simplified view if we consider the magnetic field arrangement in Figure S10 with the field strength in B2 equal and opposite that in B1 we can see that for any unscattered neutron the field integral comes to zero. However, if the neutron is scattered then the field integral in B1 is not cancelled out by B2 and will not come back to the original polarisation (P0).
By measuring the polarisation of scattered neutrons PS compared to the polarisation of the unscattered beam P0,(this allows for the correction of instrumental effects and is measured with no sample). The spin-echo lengths that can be probed using this method are determined using the following equation: Where c is a constant, L is the separation between magnets, is the inclination of the magnets, B is magnetic field strength and λ is a function of neutron wavelength 10 . Hence with a combination of these parameters we can measure the size of scattering particles in a size range up to 20μm.
The polarisation (P(z)=Ps(z)/P0(z)) of the neutrons as a function of the spin-echo length, z is given by;

( ) = ( &[(N(O)#"])
Where Σt is is the fraction of neutrons that are scattered once by a sample of thickness t and G(z) is the projection of the autocorrelation of the density distribution function of the sample 11 . For dilute spheres of radius R (volume fraction (Φ) >5%) this quantity Σt is given by; Where t is the sample thickness and λ the neutron wavelength, the scattering length density contrast ( 1 − 4 ) ! , the same as in the SANS case. Hence for any known particle size and contrast we can determine the volume fraction Φ.